Integral presentations of deviations of de la Vallee Poussin right-angled sums

Rukasov, Vladimir I.   ; Rovenska, Olga G.

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 129-137

Abstract (VO)

We investigate approximation properties of de la Vallee Poussin right-angled sums on the classes of periodic functions of several variables with a high smoothness. We obtain integral presentations of deviations of de la Vallee Poussin sums on the classes $C_{\beta ,\infty }^{m\alpha }$.

Détail

Classification : 42A10
Keywords: Right-angled sums of Vallee Poussin; integral presentations; Fourier series

Bibliographie

[1] Stepanec, A. I., Pachulia, N. L.: Multiple Fourier sums on the sets of $(\psi ,\beta )$-differentiable functions. Ukrainian Math. J. 43, 4 (1991), 545–555 (in Russian). | MR

[2] Lassuria, R. A.: Multiple Fourier sums on the sets of $\overline{\psi }$-differentiable functions. Ukrainian Math. J. 55, 7 (2003), 911–918 (in Russian). | MR

[3] Zaderey, P. V.: Integral presentations of deviations of linear means of Fourier series on the classes of differentiable periodic functions of two variables. Some problems of the theory of functions: collection of scientific works, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, 1985, 16–28 (in Russian). | MR

[4] Stepanec, A. I.: Uniform approximation by trigonometric polynomials. Nauk. Dumka, Kiev, 1981 (in Russian). | MR

[5] Stepanec, A. I.: Approximation of some classes of periodic functions two variables by Fourier sums. Ukrainian Math. J. 25, 5 (1973), 599–609 (in Russian). | MR

[6] Nikol’skii, S. M.: Approximation of the functions by trigonometric polynomials in the mean. News of Acad. of Sc. USSR 10, 3 (1946), 207–256 (in Russian). | MR

[7] Stechkin, S. B.: Estimation of the remainder of Fourier series for the differentiable functions. Works of Math. Inst. Acad. of Sc. USSR 145 (1980), 126–151 (in Russian). | MR

[8] Stepanec, A. I.: Approximation by Fourier sums of de la Poussin integrals of continuous functions. Lect. of Rus. Acad. of Sc. 373, 2 (2000), 171–173 (in Russian).

[9] Rukasov, V. I., Chaichenko, S. O.: Approximation of the classes of analytical functions by de la Vallee-Poussin sums. Ukrainian Math. J. 55, 6 (2003), 575–590. | MR

[10] Rukasov, V. I., Chaichenko, S. O.: Approximation of continuous functions by de la Vallee-Poussin operators. Ukrainian Math. J. 55, 3 (2003), 498–511. | MR

[11] Rukasov, V. I., Novikov, O. A.: Approximation of analytical functions by de la Vallee Poussin sums. Fourier series: Theory and Applications. Works of the Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, 1998, 228–241 (in Russian). | MR

[12] Stepanec A. I., Rukasov V. I., Chaichehko S. O.: Approximation by de la Vallee Poussin sums. Works of the Institute of Mathematics, Ukrainian Academy of Sciences, 68, 2007, 368 pp. (in Russian).

[13] Rukasov, V. R., Novikov, O. A., Velichko, V. E., Rovenska, O. G., Bodraya, V. I.: Approximation of the periodic functions of many variables with a high smoothness by Fourier right-angled sums. Works of the Institute of Mathematics and Mechanics, Ukrainian Academy of Sciences, 2008, 163–170 (in Russian). | MR

[14] Rukasov, V. I., Novikov, O. A., Bodraya, V. I.: Approximation of the classes of functions of two variables with a high smoothness by the right-angled linear means of Fourier series. Problems of the aproximation of the functions theory and closely related concepts. Works of the Institute of Mathematics, Ukrainian Academy of Sciences, 4, 1 (2007), 270–283 (in Russian).

[15] Stepanec, A. I.: Classification and approximation of periodic functions. Nauk. Dumka, Kiev, 1987 (in Russian). | MR